Uniform Local Weak o‐Minimality and ∗$*$‐Local Weak o‐Minimality

ABSTRACT We propose the notions of uniform local weak o‐minimality and ‐local weak o‐minimality. Local monotonicity theorems hold in definably complete locally o‐minimal structures and uniformly locally o‐minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o‐minimal structures of the second kind and for locally o‐minimal structures under the assumption called the univariate ‐continuity property. We also prove that several formulas for dimension of definable sets, which hold in definably complete locally o‐minimal structures, also hold in ‐locally weakly o‐minimal structures possessing the univariate ‐continuity property.